New identities relating wild Goppa codes

For a given support L ∈ Fn qm and a polynomial g ∈ Fqm [x] with no roots in Fqm , we prove equality between the q–aryGoppa codesΓq ( L,N(g ) ) =Γq ( L,N(g )/g ) whereN(g ) denotes thenorm of g , that is gq +···+q+1. In particular, for m = 2, that is, for a quadratic extension, we get Γq ( L,gq ) = Γq ( L,gq+1 ) . If g has roots in Fqm , then we do not necessarily have equality and we prove that the difference of the dimensions of the two codes is bounded above by the number of distinct roots of g in Fqm . These identities provide numerous code equivalences and improved designed parameters for some families of classical Goppa codes. Introduction Let Fqm /Fq be an extension of finite fields. Given an ordered n–tuple L = (α1, . . . ,αn ) ∈ Fnqm and a polynomial G ∈ Fqm [x] with no roots among the entries of L, the classical Goppa code over Fq denoted byΓq (L,G) is defined as Γq (L,G) def = { (c1, . . . ,cn )∈ F n q ∣